James Godwin’s deliriously weird puppet show, “The Flatiron Hex,” takes place in a New York only a little different from the one we know and sporadically love: “a maze of ghosts and minor gods, floating in the middle of a toxic swamp.” A postmodern assemblage of the eerie and the icky, it follows Wylie Walker, a plumber, I.T. expert and high-level shaman, as he works to protect the city from a catastrophic storm.

At the start of the play, Mr. Godwin enters the Dixon Place stage wearing a mask like a gazelle’s skull and murmuring ominously, an almost cozy entrance compared with what comes after. The plot that unfurls somehow whirls together Mickey Spillane, H. P. Lovecraft, an AppleCare employee manual and occasional gouts of blood.

Though Mr. Godwin most often plays our hero, Wylie, he also voices the other characters, and manipulates them, too. Some are dolls, some are marionettes, some are paper cutouts, some are stranger than that. Much of the staging involves overhead projectors and when the light bulb on one broke during a preview performance, Mr. Godwin had to improvise, frenetically, while a couple of technicians repaired it. “Please reset your imaginations,” Mr. Godwin said when he was ready to go on.

That’s no easy feat. “The Flatiron Hex,” directed by Tom Burnett, who is also a co-writer and the sound designer, demands that your mind’s eye operate in overdrive. There are toad gods and rat queens and animate mainframes and lascivious thumb drives and a disembodied head that haunts a train by “singing Sinatra songs and giving relationship advice.” It’s not every solo performance that makes you wonder if the usher has slipped some psilocybin into your preshow drink. “Tonight’s show is so exciting you’ll pay for a whole seat, but you’ll only use the edge,” Mr. Godwin bragged. The boast isn’t entirely idle.

There are double crosses, then triple, quadruple and quintuple ones. The plot makes very little sense and probably made even less than usual during that preview performance, as Mr. Godwin couldn’t stop inserting jokes and gripes about that blown bulb into nearly every scene. But in full light or half-light or no light at all, “The Flatiron Hex” is a dauntingly original and possibly hallucinogenic delight.